Problem: $ B = \left[\begin{array}{rr}-2 & 4 \\ 0 & 0\end{array}\right]$ $ C = \left[\begin{array}{rrr}5 & 5 & 2 \\ -1 & 4 & 5\end{array}\right]$ What is $ B C$ ?
Solution: Because $ B$ has dimensions $(2\times2)$ and $ C$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ B C = \left[\begin{array}{rr}{-2} & {4} \\ {0} & {0}\end{array}\right] \left[\begin{array}{rrr}{5} & \color{#DF0030}{5} & \color{#9D38BD}{2} \\ {-1} & \color{#DF0030}{4} & \color{#9D38BD}{5}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ B$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ B$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ B$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{-2}\cdot{5}+{4}\cdot{-1} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ B$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{-2}\cdot{5}+{4}\cdot{-1} & ? & ? \\ {0}\cdot{5}+{0}\cdot{-1} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ B$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{-2}\cdot{5}+{4}\cdot{-1} & {-2}\cdot\color{#DF0030}{5}+{4}\cdot\color{#DF0030}{4} & ? \\ {0}\cdot{5}+{0}\cdot{-1} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{-2}\cdot{5}+{4}\cdot{-1} & {-2}\cdot\color{#DF0030}{5}+{4}\cdot\color{#DF0030}{4} & {-2}\cdot\color{#9D38BD}{2}+{4}\cdot\color{#9D38BD}{5} \\ {0}\cdot{5}+{0}\cdot{-1} & {0}\cdot\color{#DF0030}{5}+{0}\cdot\color{#DF0030}{4} & {0}\cdot\color{#9D38BD}{2}+{0}\cdot\color{#9D38BD}{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-14 & 6 & 16 \\ 0 & 0 & 0\end{array}\right] $